arXiv subjects

Hironobu Naoe

Publications and source records attributed to Hironobu Naoe.

13 recordsLinked to original sources

arXiv2024

Shadow-complexity and trisection genus

The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev's shadows, which, roughly speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In this paper, we define a new version $\mathrm{sc}_{r}$ of shadow-complexity depending on an extra parameter $r\geq0$, and we investigate the relationship between this complexity and the trisection genus $g$. More explicitly, we prove an inequality $g(W) \leq 2+2\mathrm{sc}_{r}(W)$ for any closed $4$-manifold $W$ and any $r\geq1/2$. Moreover, we determine the exact values of $\mathrm{sc}_{1/2}$ for infinitely many $4$-manifolds, and also we classify all the closed $4$-manifolds with $\mathrm{sc}_{1/2}\leq1/2$.

math.GT↗

arXiv2023

The special shadow-complexity of $\#_k(S^1\times S^3)$

The special shadow-complexity is an invariant of closed $4$-manifolds defined by Costantino using Turaev's shadows. We show that for any positive integer $k$, the special shadow-complexity of the connected sum of $k$ copies of $S^1\times S^3$ is exactly $k+1$.

math.GT↗

arXiv2023

Positive flow-spines and contact 3-manifolds, II

This paper corresponds to Section 8 of arXiv:1912.05774v3 [math.GT]. The contents until Section 7 are published in Annali di Matematica Pura ed Applicata as a separate paper. In that paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.

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arXiv2023

Positive flow-spines and contact 3-manifolds

A flow-spine of a 3-manifold is a spine admitting a flow that is transverse to the spine, where the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. We say that a contact structure on a closed, connected, oriented 3-manifold is supported by a flow-spine if it has a contact form whose Reeb flow is a flow of the flow-spine. It is known by Thurston and Winkelnkemper that any open book decomposition of a closed oriented 3-manifold supports a contact structure. In this paper, we introduce a notion of positivity for flow-spines and prove that any positive flow-spine of a closed, connected, oriented 3-manifold supports a contact structure uniquely up to isotopy. The positivity condition is critical to the existence of the unique, supported contact structure, which is also proved in the paper.

math.GT↗

arXiv2023

Some lower bounds for the Kirby-Thompson invariant

Kirby and Thompson introduced a non-negative integer-valued invariant, called the Kirby-Thompson invariant, of a $4$-manifold using trisections. In this paper, we give some lower bounds for the Kirby-Thompson invariant of certain $4$-manifolds. As an application, we determine the Kirby-Thompson invariant of the spin of $L(2,1)$, which is the first example of a $4$-manifold with non-trivial Kirby-Thompson invariant. We also show that there exist $4$-manifolds with arbitrarily large Kirby-Thompson invariant.

math.GT↗

arXiv2022

Shadows of 2-knots and complexity

We introduce a new invariant for a $2$-knot in $S^4$, called the shadow-complexity, based on the theory of Turaev shadows, and we give a characterization of $2$-knots with shadow-complexity at most $1$. Specifically, we show that the unknot is the only $2$-knot with shadow-complexity $0$ and that there exist infinitely many $2$-knots with shadow-complexity $1$.

math.GT↗

arXiv2022

Presentation of the fundamental groups of complements of shadows

A shadowed polyhedron is a simple polyhedron equipped with half integers on regions, called gleams, which represents a compact, oriented, smooth 4-manifold. The polyhedron is embedded in the 4-manifold and it is called a shadow of that manifold. A subpolyhedron of a shadow represents a possibly singular subsurface in the 4-manifold. In this paper, we focus on contractible shadows obtained from the unit disk by attaching annuli along generically immersed closed curves on the disk. In this case, the 4-manifold is always a 4-ball. Milnor fibers of plane curve singularities and complexified real line arrangements can be represented in this way. We give a presentation of the fundamental group of the complement of a subpolyhedron of such a shadow in the 4-ball. The method is very similar to the Wirtinger presentation of links in knot theory.

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arXiv2020

Milnor fibration, A'Campo's divide and Turaev's shadow

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies this property and the Lefschetz fibration of this polyhedron is isomorphic to the Lefschetz fibration of the divide. Furthermore, applying the same technique to certain free divides we will show that the links of those free divides are fibered with positive monodromy.

math.GT↗

arXiv2019

Shadows of acyclic 4-manifolds with sphere boundary

In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is diffeomorphic to the standard 4-ball.

math.GT↗

arXiv2018

Four-manifolds with shadow-complexity one

We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev's shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise here all the closed oriented 4-manifolds that have complexity at most one. They are generated by a certain set of 20 blocks, that are some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb{CP}^2$ with either orientation. All the manifolds generated by these blocks are doubles. Many of these are doubles of 2-handlebodies and are hence efficiently encoded using finite presentations of groups. In contrast to the complexity zero case, in complexity one there are also plenty of doubles that are not doubles of 2-handlebodies, like for instance $\mathbb{RP}^3 \times S^1$.

math.GT↗

arXiv2017

Corks with large shadow-complexity and exotic 4-manifolds

We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an infinite family $\{(W_{n,k},W'_{n,k})\}_{k=1}^{\infty}$ of exotic pairs of $4$-manifolds with boundary whose special shadow-complexities satisfy the above inequalities. We also discuss exotic pairs with small shadow-complexity.

math.GT↗

arXiv2017

Shadows of 4-manifolds with complexity zero and polyhedral collapsing

Our purpose is to classify acyclic 4-manifolds having shadow complexity zero. In this paper, we focus on simple polyhedra and discuss this problem combinatorially. We consider a shadowed polyhedron $X$ and a simple polyhedron $X_0$ that is obtained by collapsing from $X$. Then we prove that there exists a canonical way to equip internal regions of $X_0$ with gleams so that two 4-manifolds reconstructed from $X_0$ and $X$ are diffeomorphic. We also show that any acyclic simple polyhedron whose singular set is a union of circles can collapse onto a disk. As a consequence of these results, we prove that any acyclic 4-manifold having shadow complexity zero with boundary is diffeomorphic to a 4-ball.

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arXiv2015

Mazur manifolds and corks with small shadow complexities

In this paper we find infinitely many Mazur type manifolds and corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also find such manifolds among 4-manifolds constructed from Bing's house. Our manifolds with shadow complexity one contain the Mazur manifolds $W^{\pm }(l,k)$ which were studied by Akbulut and Kirby.

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